Curves on surfaces

This is a course about very low dimensional topology, specifically about 1-manifolds immersed or embedded in 2-manifolds. Some possible topics with some suggested references are below. (links should work on campus, or off-campus, using PassKey)

Geometry of the curve complex

Gabai's Simple Loop Theorem

Intersection numbers and self-intersection numbers

Cube complexes from curves on surfaces

Automorphisms of surfaces

One of the reasons to study curves on surfaces is to understand automorphisms of surfaces and the mapping class group of a surface. Two books on this topic:

Other topics

The topics above are not the only interesting aspects of curves on surfaces. Other possibilities include:


  Date   Speaker   Topic   Reference
23 Aug Jason Intro to the course, and some hyperbolic geometry Casson and Bleiler
28 Aug Jason Some hyperbolic geometry Bruno Martelli, An introduction to Geometric Topology, especially chapter 6.
30 Aug Jason More hyperbolic geometry.
4 Sep Jason The bigon criterion.
6 Sep John Separability in surface groups Peter Scott, Subgroups of surfaces groups are almost geometric.
13 Sep Philip Minimally self-intersecting curves J. Hass and P. Scott. "Intersections of curves on surfaces." Israel J. Math. 51 (1985), no. 1-2, 90–120. MR804478
20 Sep Chaitanya Introduction to Goldman bracket M. Chas. "The Goldman bracket and the intersection of curves on surfaces." in Geometry, groups and dynamics, 73–83, Contemp. Math., 639, Amer. Math. Soc., Providence, RI, 2015. MR3379820 and "Minimal intersection of curves on surfaces," Geom. Dedicata 144 (2010), 25–60. MR250416.
27 September Jim Gabai's simple loop theorem D. Gabai. The simple loop conjecture.
16 October John The simple loop theorem for Seifert spaces J. Hass. Minimal surfaces in manifolds with S1 actions and the simple loop conjecture for Seifert fibered spaces.
Jason Manning's home page.

Last Updated 2018-10-17